The background to this question (which is not important for the actual question!) is that I'm working on something in geometric dynamics, specifically the Jacobi metric. To me it seems that one uses the kinetic energy of a dynamical system to define a pseudo-Riemannian metric on a manifold. However it is not clear to me whether this is actually always possible and if so how.
My question is whether a positive definite quadratic form (which I assume reasonably defined kinetic energies to be) induce positive nondegenerate or at least nondegenerate inner products. (For those in the know about geometrical mechanics this means whether a kinetic energy does indeed define a (pseudo-)Riemannian metric on a manifold).
Let $V$ be a vector space and let $S$ be a quadratic form on $V$ such that
- $S(v)=0$ iff $v=0$
- $S(v)>0$ for all $v\neq 0$
This defines a symmetric inner product, call it $g$, on $V$ as follows $g(v,w)=\frac{1}{2}(S(v+w)-S(v)-S(w))$
Under what circumstances is $g$ positive and/or nondegenerate?